How can I understand linear functions and their matrices in polynomials?

In summary, the function is linear if and only if it satisfies the following conditions: -The derivative is dimensionless (meaning it has the same number of dimensions as the original function) -The derivative is closed under vector addition and scalar multiplication.
  • #1
mrroboto
35
0
I should just give math up, I don't understand this at all. It seems like for a) the function could be squared, and other than that it doesn't make any sense.


Let V = {p element of R[x] | deg(p) <=3} be the vector space of all polynomials of degree 3 or less.

a) Explain why the derivative d/dx: V -> V is a linear function

b) Give the matrix for d2/dx2 in the basis {1,x,x^2, x^3} for V

c) Give the matrix for the third derivative d2/dx2: V->V using the same basis

d) Give the matrix for the third derivative d3/dx3: V->V using the same basis

e) Give a basis for ker(d2/dx2)

f) what is the matrix for the linear map (d/dx + 4(d3/dx)): V->V

g) Let T: R[x] ->R[x] be the function T(p(x))= integral from -2x to 2 of p(t)dt

Explain why T is an element of L(R[x])
 
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  • #2
OK, let's start with a)
What does it mean for a function to be linear?
What is the formula you have to check, that needs to be true for a linear function f?
 
  • #3
That is is closed under vector addition and scalar multiplication? In this case, the derivative's dimension will always be a square (or less), and it is closed under addition because added two squares will always be within a cube, and it is closed under multiplication because any constant times a square will still be within a cube.
 
  • #4
I think your problem is that you don't learn the definitions well enough! In mathematics, all definitions are "working" definitions- you use the precise words of the definition in proofs and problems.

My point is that you were asked, "What does it mean for a function to be linear?" by Compuchip and answered, "That is is closed under vector addition and scalar multiplication? " That has nothing to do with functions at all- that is a definition of "subspace". A function f(x) on a vector space is linear if and only if f(u+ v)= f(u)+ f(v) and f(av)= af(v) for any vectors u and v and scalar a. If p(x) and q(x) are in your set of polynomials, what is d(p+q)/dx? If, also, a is a number, what is d(ap(x))/dx?

A standard way of finding the matrix representing a linear transformation in a given basis is to apply the linear transformation to each basis vector in turn. The coefficients of the result, written in that basis, form the colums of the matrix.
 
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1. What are complex numbers?

Complex numbers are numbers that contain both a real and imaginary component. They are written in the form a + bi, where a is the real part and bi is the imaginary part with i being the square root of -1.

2. How are complex numbers represented in a matrix?

Complex numbers can be represented in a matrix by arranging the real and imaginary parts into rows and columns. For example, the complex number 2 + 3i can be represented as [2 3; -3 2] in a 2x2 matrix.

3. What operations can be performed on matrices of complex numbers?

Just like with real numbers, matrices of complex numbers can be added, subtracted, multiplied, and divided. However, when multiplying matrices of complex numbers, the order of multiplication matters.

4. What is the conjugate of a complex number?

The conjugate of a complex number a + bi is the number a - bi. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

5. How are complex numbers used in science and mathematics?

Complex numbers are used in various fields of science and mathematics, such as engineering, physics, and signal processing. They provide a way to represent and manipulate quantities that have both magnitude and direction, such as alternating currents in electrical engineering.

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